BRIDGES

Rekha R. Thomas, University of Washington

Rekha Thomas is a Walker Family Endowed Professor of Mathematics at the University of Washington in Seattle. She received her Ph.D. in Operations Research from Cornell University in 1994, followed by postdocs at Yale University and ZIB, Berlin. Her research interests are in Optimization, Combinatorics and Applied Algebraic Geometry.

Lecture Series: Three Problems on Graphs
Abstract: In this talk series, I will introduce three problems on graphs that turn out to have intimate connections to combinatorics, discrete and convex geometry, and optimization.  The first is about designs in graphs which are subsets of vertices over which functions on graphs have the same average as over the whole graph. The second topic is about sparsifiers of a graph that preserve the low end of the spectral information of the graph. The final topic is about graphs that are rigid in the sense that allowing weights on their edges will not increase the second smallest eigenvalue of the Laplacian or decrease the largest eigenvalue. All of these problems have connections to applications and bring up new research questions.

These talks are based on the following papers:
https://arxiv.org/pdf/2204.01873
https://arxiv.org/pdf/2306.06204
https://arxiv.org/abs/2402.11758

Lecture Notes
Problem Session

Kim Ruane, Tufts University

Kim is a Professor in the Department of Mathematics at Tufts University whose research is in Geometric Group Theory. Kim is a first-generation college student with a somewhat colorful journey through the educational system. Upon hearing that Kim wanted to get a Ph.D. in math, her high school guidance counselor replied, “well if you’re not in jail, maybe…”. With solid mentoring and support from a variety of people as well as some good old-fashioned hard work, she was able to realize her dream of becoming a college professor who gets to do math and teach math for a living. Now that she is an established Professor, she prefers collaborative mathematics, as well as teaching and mentoring at all levels. Her current passion is prison education. She regularly teaches math courses inside a medium security prison as part of the Tufts University Prison Initiative through Tisch College (TUPIT). The program allows students to work towards an Associates and/or BA degree while incarcerated. This has been the most rewarding and eye-opening experience in terms what it means to teach mathematics.

Lecture Series: Infinite Groups as Geometric Objects
Abstract: A finitely generated, infinite group can be viewed as a geometric object in two ways: via a Cayley graph for the group or by realizing the group as a group of isometries of a more robust (non-compact) metric space. In this way, the algebraic properties of the group are related to the geometric properties of a space. This viewpoint goes back to the work of Dehn in the 1920’s on surface groups, but was popularized by Gromov, Cannon, Thurston and others in the late 1980’s with the definition of Gromov hyperbolic group. The metric spaces we work with can be compactified by adding a boundary to them. My work in the area has centered on the study of boundaries for different kinds of group. In the first lecture, we will give an introduction to studying groups as geometric objects. In the second, we will study lots of examples and discuss boundaries. In the third lecture, I would like to discuss the Cannon Conjecture since it is the conjecture that motivated me to study boundaries when I was a graduate student.

Problem Session 1
Problem Session 2

Ila Varma, University of Toronto

Ila Varma is an Assistant Professor at the University of Toronto. She graduated from Princeton University in 2015 with a PhD after obtaining a MSc from Leiden University and a BS from Caltech. Her research is in number theory, with a current focus in arithmetic statistics. Ila is heavily involved in outreach, unlearning inequitable academic and teaching practices, and championing marginalised voices, ideas, and individuals within the mathematics community. In line with these values, she is the founder of a variety of mathematical activities across the globe, including PROMYS India, a free program for high school students, which takes place at IISc Bangalore and the Equity Forum, a seminar for all members of the mathematical community at University of Toronto.

Lecture Series: Counting Number Fields and Predicting Asymptotics
Abstract: A foundational question in number theory, specifically in the subfield called arithmetic statistics, is: How many number fields are there? Number fields are  objects that are utilized heavily in the study of number theory and algebraic geometry. Typically, we are introduced to number fields in Galois theory classes. More precisely, number fields are vector spaces over $\mathbb{Q}$ that include both the rational numbers and the roots of a fixed polynomial $f(x)$ under the operations of addition and multiplication. Like other fields, every nonzero element of a number field has a multiplicative inverse. If we allow ourselves to vary over polynomials of a fixed degree $n$, we can refine the question to the way number theorists like to study it: How many number fields of a fixed dimension $n$ are there? And, if we filter the family of polynomials not only by degree but on the types of symmetries the roots have dictated by what's known as the Galois group, we arrive at the questions surrounding Malle's Conjecture: precisely, how does the count of number fields of degree n whose normal closure has Galois group G grow as their discriminants tend to infinity? We will discuss the history of this question including its connection to the inverse Galois problem and take a closer look at the story in the case that $n = 2,3,4$, i.e. the counts of quadratic, cubic, and quartic fields.

Lecture Notes 1
Lecture Notes 2
Problem Session